The "enhanced spectrum" of an image g[.] is a function h[.] of wave-number u obtained by a sequence of operations on the power spectral density of g[.]. The main properties and the available theorems on the correspondence between spectrum enhancement and spatial differentiation, of either integer or fractional order, are stated. In order to apply the enhanced spectrum to image classification, one has to go, by interpolation, from h[.] to a polynomial q[.]. The graph of q[.] provides the set of morphological descriptors of the original image, suitable for submission to a multivariate statistical classifier. Since q[.] depends on an n-tuple, Ψ, of parameters which control image pre-processing, spectrum enhancement and interpolation, then one can train the classifier by tuning Ψ. In fact, classifier training is more articulated and relies on a "design", whereby different training sets are processed. The best performing n-tuple, Ψ*, is selected by maximizing a "design-wide" figure of merit. Next one can apply the trained classifier to recognize new images. A recent application to materials science is summarized.